On a conjecture of Graham on the p-divisibility of central binomial coefficients

TL;DR

This paper proves that for any fixed number of large primes, there are infinitely many binomial coefficients with controlled prime divisibility, and relates this to Graham's conjecture on the coprimality of central binomial coefficients with 105.

Contribution

It establishes the existence of infinitely many integers where the divisibility of central binomial coefficients by large primes is minimal, extending Kummer's bounds and connecting to Graham's conjecture.

Findings

Existence of infinitely many n with low prime multiplicity divisibility

Bound on prime divisibility multiplicity relative to log n and p_j

Connection to Graham's conjecture on coprimality with 105

Abstract

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is $1+\log n / \log p_j$; and our theorem shows that for every $\varepsilon > 0$, $r \geq 1$, and any sufficiently large primes $p_1,...,p_r > p_0(\varepsilon,r)$, we can find integers $n$ where for $j=1,...,r$, $p_j$ divides ${2n \choose n}$ with multiplicity at most $\varepsilon \log n/\log p_j$. We connect this result to a famous conjecture by R. L. Graham on whether there are infinitely many integers $n$ such that ${2n \choose n}$ is coprime to $105$.